6,725 research outputs found
Canonical Proof nets for Classical Logic
Proof nets provide abstract counterparts to sequent proofs modulo rule
permutations; the idea being that if two proofs have the same underlying
proof-net, they are in essence the same proof. Providing a convincing proof-net
counterpart to proofs in the classical sequent calculus is thus an important
step in understanding classical sequent calculus proofs. By convincing, we mean
that (a) there should be a canonical function from sequent proofs to proof
nets, (b) it should be possible to check the correctness of a net in polynomial
time, (c) every correct net should be obtainable from a sequent calculus proof,
and (d) there should be a cut-elimination procedure which preserves
correctness. Previous attempts to give proof-net-like objects for propositional
classical logic have failed at least one of the above conditions. In [23], the
author presented a calculus of proof nets (expansion nets) satisfying (a) and
(b); the paper defined a sequent calculus corresponding to expansion nets but
gave no explicit demonstration of (c). That sequent calculus, called LK\ast in
this paper, is a novel one-sided sequent calculus with both additively and
multiplicatively formulated disjunction rules. In this paper (a self-contained
extended version of [23]), we give a full proof of (c) for expansion nets with
respect to LK\ast, and in addition give a cut-elimination procedure internal to
expansion nets - this makes expansion nets the first notion of proof-net for
classical logic satisfying all four criteria.Comment: Accepted for publication in APAL (Special issue, Classical Logic and
Computation
Elliptic nets and elliptic curves
An elliptic divisibility sequence is an integer recurrence sequence
associated to an elliptic curve over the rationals together with a rational
point on that curve. In this paper we present a higher-dimensional analogue
over arbitrary base fields. Suppose E is an elliptic curve over a field K, and
P_1, ..., P_n are points on E defined over K. To this information we associate
an n-dimensional array of values in K satisfying a nonlinear recurrence
relation. Arrays satisfying this relation are called elliptic nets. We
demonstrate an explicit bijection between the set of elliptic nets and the set
of elliptic curves with specified points. We also obtain
Laurentness/integrality results for elliptic nets.Comment: 34 pages; several minor errors/typos corrected in v
Expansion Trees with Cut
Herbrand's theorem is one of the most fundamental insights in logic. From the
syntactic point of view it suggests a compact representation of proofs in
classical first- and higher-order logic by recording the information which
instances have been chosen for which quantifiers, known in the literature as
expansion trees.
Such a representation is inherently analytic and hence corresponds to a
cut-free sequent calculus proof. Recently several extensions of such proof
representations to proofs with cut have been proposed. These extensions are
based on graphical formalisms similar to proof nets and are limited to prenex
formulas.
In this paper we present a new approach that directly extends expansion trees
by cuts and covers also non-prenex formulas. We describe a cut-elimination
procedure for our expansion trees with cut that is based on the natural
reduction steps. We prove that it is weakly normalizing using methods from the
epsilon-calculus
Relative Entropy in CFT
By using Araki's relative entropy, Lieb's convexity and the theory of
singular integrals, we compute the mutual information associated with free
fermions, and we deduce many results about entropies for chiral CFT's which are
embedded into free fermions, and their extensions. Such relative entropies in
CFT are here computed explicitly for the first time in a mathematical rigorous
way. Our results agree with previous computations by physicists based on
heuristic arguments; in addition we uncover a surprising connection with the
theory of subfactors, in particular by showing that a certain duality, which is
argued to be true on physical grounds, is in fact violated if the global
dimension of the conformal net is greater than Comment: 31 page
Relative Entropy in CFT
By using Araki's relative entropy, Lieb's convexity and the theory of
singular integrals, we compute the mutual information associated with free
fermions, and we deduce many results about entropies for chiral CFT's which are
embedded into free fermions, and their extensions. Such relative entropies in
CFT are here computed explicitly for the first time in a mathematical rigorous
way. Our results agree with previous computations by physicists based on
heuristic arguments; in addition we uncover a surprising connection with the
theory of subfactors, in particular by showing that a certain duality, which is
argued to be true on physical grounds, is in fact violated if the global
dimension of the conformal net is greater than Comment: 31 page
Estimating operator norms using covering nets
We present several polynomial- and quasipolynomial-time approximation schemes
for a large class of generalized operator norms. Special cases include the
norm of matrices for , the support function of the set of
separable quantum states, finding the least noisy output of
entanglement-breaking quantum channels, and approximating the injective tensor
norm for a map between two Banach spaces whose factorization norm through
is bounded.
These reproduce and in some cases improve upon the performance of previous
algorithms by Brand\~ao-Christandl-Yard and followup work, which were based on
the Sum-of-Squares hierarchy and whose analysis used techniques from quantum
information such as the monogamy principle of entanglement. Our algorithms, by
contrast, are based on brute force enumeration over carefully chosen covering
nets. These have the advantage of using less memory, having much simpler proofs
and giving new geometric insights into the problem. Net-based algorithms for
similar problems were also presented by Shi-Wu and Barak-Kelner-Steurer, but in
each case with a run-time that is exponential in the rank of some matrix. We
achieve polynomial or quasipolynomial runtimes by using the much smaller nets
that exist in spaces. This principle has been used in learning theory,
where it is known as Maurey's empirical method.Comment: 24 page
Stable quantum systems in anti-de Sitter space: Causality, independence and spectral properties
If a state is passive for uniformly accelerated observers in n-dimensional
anti-de Sitter space-time (i.e. cannot be used by them to operate a perpetuum
mobile), they will (a) register a universal value of the Unruh temperature, (b)
discover a PCT symmetry, and (c) find that observables in complementary
wedge-shaped regions necessarily commute with each other in this state. The
stability properties of such a passive state induce a "geodesic causal
structure" on AdS and concommitant locality relations. It is shown that
observables in these complementary wedge-shaped regions fulfill strong
additional independence conditions. In two-dimensional AdS these even suffice
to enable the derivation of a nontrivial, local, covariant net indexed by
bounded spacetime regions. All these results are model-independent and hold in
any theory which is compatible with a weak notion of space-time localization.
Examples are provided of models satisfying the hypotheses of these theorems.Comment: 27 pages, 1 figure: dedicated to Jacques Bros on the occasion of his
70th birthday. Revised version: typos corrected; as to appear in J. Math.
Phy
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